WHere will the lim be shifting from the left( - ) or from the right ( + )? and why?

Factor, your idea was correct.

Now when we approach the limit from the right, that means,
But then,
And the function is undefined.
(Radical is a negative number!)

So the limit (the regular limit) does not exist.
However, the one-handed limit, might. We already shown that the right-handed does not exists. Let us see the left-handed.

When we approach the limit from the left, on some open interval where (a really small number). Because we do not care what happens far away one what happens really close to the point.
In that case,
Thus, the radical is defined.
And furthermore we can factor,

Well, I don't know, but I think so You just have to choose whether x > 2 or x < 2 I guess ... do you have an example of when it doesn't work?

I don't have an example of where it doesn't work, I was simply speculating. I was just trying to be careful since the derivative of doesn't exist at x = 2 (because the function doesn't exist for x < 2.) Apparently as long as the limit of the derivative exists, L'Hopital is usable, if I'm understanding TPH correctly.

First I would like to mention square roots of of negative numbers are undefined. Note, it is mathematically incorrect to say (I know that is what they teach in high schools, but it is still wrong). I have already discussed that on the forum. Next, all the L'Hopitals rule was only proven for real differenciable functions (at least I think). Finally this problem it self is real, implies that the function the poster was using mapped the real numbers into the real numbers.